Diffusion approximation for an age-structured population
Consider a population where each individual either gives birth to a single offspring or dies with constant and equal rates \(\lambda\). The population size process \(N = (N_t)_{t \geq 0}\) is a linear, critical birth-and-death process. The authors study the distribution of age among the members of the population. The age distribution is described by the measure-valued process \(X = (X_t)_{t \geq 0}\) such that \(X_t ([x_1, x_2])\) = number of individuals present at time \(t\) with ages in the interval \([x_1, x_2]\). It is well-known that the small mass and short life limit of the process \(N\) is the so-called ``continuous state branching process'' with parameter \(\lambda\). The purpose of the paper is to show that an analogous result holds for the age process: If \(X^j\), \(j = 1,2, \ldots,\) are i.i.d. copies of the age process, then the measure-valued process \(({1 \over n} \sum^n_{j = 1} X^j_{nt})_{t \geq 0}\) converges weakly as \(n \to \infty\), the limit being the process \((\overline N_t \mu_\lambda)_{t \geq 0}\), where \(\overline N\) is the continuous state branching process and \(\mu_\lambda\) is the exponential distribution with parameter \(\lambda\). Thus, in the diffusion limit the size of the age groups behaves like the total population size multiplied by an exponential weight; e.g. the diffusion limit of the individuals of age exceeding \(a\) at time \(t\) is given by \(e^{- \lambda a} \overline N_t\).
scientific article; zbMATH DE number 780985